Tight frames and their symmetries
Richard Vale and Shayne Waldron
Abstract:
The aim of this paper is to investigate symmetry properties of tight frames,
with a view to constructing tight frames of orthogonal polynomials
in several variables which share the symmetries of the weight function,
and other similar applications.
This is achieved by using representation theory to
give methods for constructing tight frames as orbits of groups of unitary
transformations acting on a given finite-dimensional Hilbert space.
Along the way, we show that a tight frame is determined by its Gram matrix
and discuss how the symmetries of a tight frame are related to its Gram matrix.
We also give a complete classification of those tight frames which arise as
orbits of an abelian group of symmetries.
Keywords:
Tight frames,
isometric tight frames,
Gram matrix,
multivariate orthogonal polynomials,
symmetry groups,
harmonic frames,
representation theory,
wavelets
Math Review Classification:
Primary 05B20, 33C50, 20C15, 42C15 ;
Secondary 52B15, 42C40i
Length: 30 pages
Last Updated: 2 February 2004
Availability: